2024 Students Enrolled in or before 2015 School of Science Mathematics
Exercises in Geometry A
- Academic unit or major
- Mathematics
- Instructor(s)
- Hisaaki Endo / Satoshi Nakamura
- Class Format
- Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-8 Tue / 5-8 Tue
- Class
- -
- Course Code
- ZUA.B204
- Number of credits
- 020
- Course offered
- 2024
- Offered quarter
- 3-4Q
- Syllabus updated
- Mar 17, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This course is an exercise session for the lecture course `Set and Topology II' (ZUA.B203). The materials for exercise are chosen from that course.
Course description and aims
Students are expected to
・Understand various equivalent definitions of topology
・Understand that continuity of maps between topological spaces is described in terms of topology
・Understand various kinds of topologies that naturally arise under various settings
・Understand various separation axioms, with various examples
・Be able to prove basic properties of connected and compact spaces
・Learn a lot of basic examples of compact/ non-compact and connected/disconnected spaces
・Understand basic properties of complete metric spaces and examples
Keywords
topology and topological space, neighborhood, first countability, second countability, continuous mapping, induced topology, separation axioms, compact space, connected spaces, path-connectedness, completeness of a metric space
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Students are given exercise problems related to what is taught in the course "Set and Topology II"
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | discussion session on the following materials: topology and topological space | Details will be provided during each class session |
| Class 2 | discussion session on the following materials: open basis, system of neighborhoods, second countability | Details will be provided during each class session |
| Class 3 | discussion session on the following materials: fundamental system of neighborhoods, first countability | Details will be provided during each class session |
| Class 4 | discussion session on the following materials: continuous map, homeomorphism | Details will be provided during each class session |
| Class 5 | discussion session on the following materials: relative topology, product topology | Details will be provided during each class session |
| Class 6 | discussion session on the following materials: quotient topology, induced topology | Details will be provided during each class session |
| Class 7 | discussion session on the following materials: Hausdorff space, normal space | Details will be provided during each class session |
| Class 8 | discussion session on the following materials: separation axioms and continuous functions | Details will be provided during each class session |
| Class 9 | discussion session on the following materials: connectedness of a topological space | Details will be provided during each class session |
| Class 10 | discussion session on the following materials: path-connectedness of a topological space | Details will be provided during each class session |
| Class 11 | discussion session on the following materials: compactness of a topological space | Details will be provided during each class session |
| Class 12 | discussion session on the following materials: properties of a compact space | Details will be provided during each class session |
| Class 13 | discussion session on the following materials: completeness of metric spaces | Details will be provided during each class session |
| Class 14 | discussion session on the following materials: topological properties of metric spaces | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
Evaluation methods and criteria
brief exam (about 30%), oral presentation for exercise problems (about 70%)
Related courses
- ZUA.B203 ： Set and Topology II
- MTH.B203 ： Introduction to Topology III
- MTH.B204 ： Introduction to Topology IV
Prerequisites
Students are expected to have passed [Calculus I / Recitation], Calculus II + Recitation, [Linear Algebra I / Recitation] and Linear Algebra II + Recitation.
Strongly recommended to take ZUA.B203 ： Set and Topology II (if not passed yet) at the same time
Other
T2SCHOLA will be used.